Helmholtz’ and Longuet-Higgins’ Theories of Consonance and Harmony

The question of what constitutes musical experience and understanding is a very ancient one, like many important questions about the mind. The answers that have been offered over the years since the question was first posed have depended on the notion of mechanism that has been available as a metaphor for the mind. For Aristotle, and for the Pythagoreans, the explanation of the musical faculty lay in the mathematics of integer ratios and the physics of simply vibrating strings. Helmholtz was able to draw upon nineteenth century physics, for a more properly mechanistic and complete explanation of the phenomenon of consonance. For him, a mechanism was a physical device such as a real resonator or oscillator. The principal tool that we have available, beyond those that Aristotle and Helmholtz knew of, is the computer. Of course, it is often the algorithm that the computer executes that is of interest, rather than the computer itself, since for many interesting cases we can state the algorithm independently of any particular machine. However, the idea of an algorithm is not in itself novel. Algorithms (such as Euclid’s algorithm) were known to Helmholtz. It is the computer which transforms the notion of an algorithm from a procedure that needs a person to execute it to the status of a mechanism or explanation.